1. IntroductionWith outstanding chemical and mechanical properties, group-IV nitrides M3N4 (
, Ge, Sn, etc.) are considered as the most promising technological ceramics used in high-temperature fields.[1,2] The high fracture toughness, unique physical property and good thermal shock resistance make Si3N4 a desirable material for hostile environments.[3] The Si3N4 nitrides can also be used as wide-gap semiconductors, laser diodes and etch masks due to their high hardnesses and good thermal stabilities.[4,5] Under ambient conditions, there are two well-established polymorphs of Si3N4, namely the hexagonal α (
and β (
phases, where the β configuration is the low-temperature phase and α-Si3N4 is often recognized as a metastable phase.[6] Several decades ago, the α- and β-Si3N4 were the only two phases known to exist.[7,8] Through the diamond anvil cell technique, the first work gave a temperature of 2000 K and a pressure of 15 GPa needed to synthesis the spinel γ phase (Fd-3m).[9] Several polymorphs of Si3N4 have been reported so far, including the still ambiguous phases {willemite-II-Si3N4 (I-43d),[10] δ-Si3N4(P3),[7] post-spinel-Si3N4 (BBMM),[11]
and
’-Si3N4(P3),[7] etc}.
The group IV nitride usually displays desirable properties including enhanced wear resistance in Si3N4, unique catalytic activity in Ge3N4, and tunable band gap in Sn3N4.[12] Due, in particular, to its outstanding heat conductivity, electro–optical and dielectric properties, germanium nitride (Ge3N4) is a prospective compound for applications in photocatalysts, optical fibers, and engineering ceramics. It has been considered for applications as passivation layers, as possible negative electrodes, and as components of read-and-write memories.[13,14] Since carriers in Ge have higher mobilities compared with those in Si, the Ge-based field-effect-transistor has attracted a great deal of attention as a high-performance device. Sn3N4 is of special interest due to its electro–chromic and semiconducting properties. From an indentation technique, γ-Sn3N4 was found to be much softer and more plastic than its former congeners γ-Si3N4 and γ-Ge3N4.[15] The existence of α- and β-Ge3N4 has been identified in the synthesized nanobelts.[16] Progress has been rapid, and now the dense nitrides γ-M3N4 (M=Si, Ge, Sn) have already been synthesized and start to be commercially available.[17] It is worthy of note that Sn3N4 is far less studied than its silicon and germanium counterparts, and the investigation remains ongoing.
Nowadays, Xu et al.[7] and Kuwabara et al.[8] proposed calculations for the phonon dispersions, density of states, thermal properties and phase stabilities of α-, β-, and γ-Si3N4 through the VASP code. Gao et al.[18] and Luo et al.[19] have investigated the band structures, elastic moduli, thermodynamic properties and phase transitions of α-, β-, and γ-Ge3N4 using the plane-wave pseudo-potential method. The electronic and optoelectronic properties of α-, β-, and γ-M3N4 (M= Si, Ge, Sn) have been analysed by Caskey et al.[12] Detailed studies of the electronic, optical and thermodynamic properties of these nitrides have already been introduced in our previous works.[19,20] In 2015, Cui et al.[21] explored three new polymorphs (tetragonal t-Si3N4, monoclinic m-Si3N4 and orthorhombic o-Si3N4) using the CALYPSO code.[22] Up to now, the fundamental properties of these compounds are still under development. For the purpose of structural consistency, we chose to describe tetragonal, monoclinic and orthorhombic Ge3N4 and Sn3N4, since α-, β-, and γ-Ge3N4/Sn3N4 (similar to silicon nitrides) had already been discovered. Consider the fact that the electronic structures between Ge/Sn and Si are quite similar, the t-, m-, and o-Ge3N4/Sn3N4 may also exist. The aim of this work is to predict the structural, electronic, optical and thermodynamic properties of t-, m-, and o-M3N4 (M=Si, Ge, Sn). We hope that our theoretical prediction will motivate future studies.
2. Model and methodsThe optimized structures of Si3N4, Ge3N4, and Sn3N4 can be found in Refs. [23] and [24]. In this work, theoretical calculations were performed in the framework of density functional theory (DFT) using the plane-wave pseudo-potential (PW-PP) method,[25] where the Perdew–Burke–Ernzerhof (PBE) functional[26] with generalized gradient approximation (GGA) was adopted to describe the exchange-correlation potentials.[27] The interactions between valence electrons and ionic cores are modeled by the Vanderbilt-type ultrasoft pseudo-potentials.[28] The electronic wave function is expanded in a plane wave basis set with a well converged cutoff energy of 500 eV for all cases. The Brillouin zones were sampled with the Monkhorst–Pack k-points[29] of 12× 12× 8, 8× 18× 12, and 11× 7× 10 for the tetragonal, monoclinic and orthorhombic phases, respectively. Pseudo-atomic valence states had been set as Si-3s23p2, Ge-4s24p2, Sn-5s25p2, and N-2s22p3. For the electronic structure calculations, the empty bands were chosen to be 53. The pristine lattice structures (including atomic positions) were relaxed through the Broyden–Fletcher–Goldfarb–Shanno (BFGS) scheme[30] with the following tolerances: energy difference per atom, maximum Hellmann–Feynman force, maximum ionic displacement, and maximum stress less than
, 0.01 eV/Å, 0.001 Å, and 0.02 GPa, respectively. The electronic, optical and thermodynamic properties are predicted based on the optimized structures.
In the full-electronic quasi-harmonic approximation (QHA) scheme,[31] the non-equilibrium Gibbs free energy
can be written as
where
is the total energy,
P represents the external pressure,
V is the cell volume, and
represents the reminder of the polynomial (magnetic, configurational, etc.).
[31]
and
are the contributions to the Helmholtz free energy from lattice vibrations and mobile electrons, respectively.
where
is the Fermi level,
Boltzmann’s constant,
the number of valence electrons,
T the temperature,
ωj the vibrational frequency,
n the number of atoms, and
N the number of cells. Thermal properties such as heat capacity
Cv,
Cp, isothermal bulk modulus
B, internal energy
U, thermal expansion coefficient
α, Grüneisen parameter
γ, and entropy
S can be derived from Eq. (
1)
where
V is the volume,
P the pressure,
θ the Debye temperature, and
the Debye integral, respectively. The detailed expressions of QHA can be found in Refs. [
31] and [
32]. The mobile electrons can be ignored in semiconductors.
3. Results and discussions3.1. Crystal structuresAs a first step, the M3N4 (
, Ge, Sn) crystals have been optimized. The results are listed in Table 1. It is shown that the calculated lattice constants a, b, c and elastic moduli B, G, E of Si3N4 are consistent with the literature,[21] while the maximum relative errors are only 0.9% and 8.5% for the lattice constants and elastic moduli, respectively (within the general accuracy of GGA, i.e., 1%
3% for lattice constants and 1%
10% for elastic moduli). The bulk moduli B are greater than the shear moduli G, which reflects that the principal parameter limiting the stabilities of t-, m-, and o-
(M=Si, Ge, Sn) is the shear modulus.[33] Since B is a measure of rigidity, we conclude that the rigidity of the orthorhombic phase is the highest among these polymorphs. Young’s modulus E follows the sequences
and
. In Table 1, it is found that all nitrides show semiconducting characteristics. The direct band gap enables the direct transitions of electrons, which is beneficial to the opto–electrical properties of t-Ge3N4, m-Si3N4, and o-Ge3N4. The remaining compounds show indirect band gaps. Si3N4 are wide band-gap (
) semiconductors while Ge3N4 and Sn3N4 belong to narrow band-gap (
) semiconductors. Although Cui et al.[21] found that m-Si3N4 belongs to direct band gap semiconductor (
); our result shows that m-Si3N4 has an
direct band gap.
Table 1.
Table 1.
 Table 1.
Calculated equilibrium structural parameters: lattice constants a, b,c, bulk modulus B, shear modulus G, Young’s modulus E, and band gap
for t-, m-, and o-M3N4 (M =Si, Ge, Sn).
.
|
a/nm |
b/nm |
c/nm |
B/GPa |
G/GPa |
E/GPa |
/GPa |
/eV |
| t-Ge3N4 |
0.4410 |
0.4410 |
0.8835 |
146.8 |
86.8 |
217.7 |
194.5 |
1.76 (Γ–Γ) |
| m-Ge3N4 |
1.0215 |
0.3103 |
0.7928 |
124.1 |
72.8 |
182.8 |
126.9 |
1.85 (A–Γ) |
| o-Ge3N4 |
0.5256 |
1.0074 |
0.5317 |
203.4 |
122.0 |
305.1 |
198.4 |
1.35 (Γ–Γ) |
| t-Sn3N4 |
0.4930 |
0.4930 |
0.9961 |
84.9 |
38.6 |
100.5 |
85.8 |
0.22 (M–Γ) |
| m-Sn3N4 |
1.1473 |
0.3487 |
0.8891 |
74.4 |
33.5 |
87.3 |
83.3 |
0.13 (E–Γ) |
| o-Sn3N4 |
0.5845 |
1.1164 |
0.5904 |
150.6 |
80.4 |
204.7 |
140.7 |
0.16 (Γ–Y) |
| t-Si3N4 |
0.4166 |
0.4166 |
0.8249 |
194.4 |
126.1 |
311.0 |
195.2 |
3.04 (M–Γ) |
| m-Si3N4 |
0.9603 |
0.2932 |
0.7419 |
165.4 |
104.3 |
258.5 |
171.1 |
3.77 (A–A) |
| o-Si3N4 |
0.4937 |
0.9426 |
0.4948 |
261.5 |
178.6 |
436.4 |
265.7 |
3.21 (Γ–Y) |
| t-Si3N4[21] |
0.4131 |
0.4131 |
0.8168 |
193.8 |
134.9 |
328.4 |
– |
3.15 (M–Γ) |
| m-Si3N4[21] |
0.9512 |
0.2904 |
0.7353 |
164.9 |
113.2 |
276.3 |
– |
3.90 (Γ–Γ) |
| o-Si3N4[21] |
0.4901 |
0.9356 |
0.4910 |
258.1 |
185.7 |
449.3 |
– |
3.36 (Γ–Y) |
| Table 1.
Calculated equilibrium structural parameters: lattice constants a, b,c, bulk modulus B, shear modulus G, Young’s modulus E, and band gap
for t-, m-, and o-M3N4 (M =Si, Ge, Sn).
. |
It is worth noting that DFT + GGA usually underestimate the band gaps of nitrides, hence the real band gaps of M3N4 (M=Si, Ge, Sn) would be wider than the predicted values. In Table 1, the
is the bulk modulus obtained by the quasi-harmonic approximation (i.e., QHA bulk modulus), while B represents the first-principles PW–PP bulk modulus. For most cases, the
and B agree well with each other. Frankly speaking, the reason for the discrepancy between
and B (for t-Ge3N4) is still unclear. This needs to be testified by future expertiments. On one hand, we have thoroughly studied the elastic constants
of t-, m-, and o-Si3N4/Ge3N4 in our previous works.[23,24] On the other hand, our calculated lattice constants, elastic constants, bulk moduli and thermal properties of α-Si3N4,β-Si3N4,γ-Si3N4, β-Ge3N4, and γ-Ge3N4 are in agreement with the experimental data.[19,34,35] These facts partially demonstrate that our results given in this work are reliable but need to be verified by future experiments.
3.2. Electronic propertiesThe total density of states (TDOS) and partial-wave density of states (PDOS) projected onto different atoms, obtained, respectively, with single M3N4(M=Si, Ge, Sn) unit (7 atoms), are plotted in Fig. 1. At first glance, it appears that the TDOSs of t-M3N4 (M=Si, Ge, Sn) drawn in Fig. 1(a) differ very little. It is clear that the DOS at Fermi level
is dominated by the contribution of N-2p orbital in combination with the weak contributions of M-s, p states. The valence band (VB) can be separated into two parts: i.e. the lower region and the upper region. The lower VB region (from about −18 eV to −12 eV) consists mainly of N-2s band hybridized with small contributions from M-s and M-p orbitals. The upper VB region from −9 eV to 0 eV is dominated by the M-s band, but hybridization occurs between the N-2p and M-p states. There are five well-resolved peaks that appear in the upper VB regions of t-M3N4(M=Si, Ge, Sn). The conduction band (CB) DOS is also rich in structures while the Si and Sn elements have comparable contributions. The Ge contribution to CB is significant. In Figs. 1(b),1(c), and 1(d), PDOSs of t-Si3N4, t-Ge3N4, and t-Sn3N4 show the same variation with energy. In the PDOS of t-Ge3N4, the lower VB region is dominated by s electrons of N and Ge, and the upper VB region is mainly composed of N-2p states (unlike t-Si3N4/Sn3N4). Follow the sequence
, the peaks of lower VB-DOSs increase, reflecting the enhanced M–N hybridization. The VB-DOSs of Si3N4 and Ge3N4 are broader than that of Sn3N4 by approximately 2.4 eV and 2.2 eV, respectively. This is caused by the Sn atom because its s-electrons are located near the bottom of the VB, which indicates that Sn3N4 are the most stable compounds among the three nitrides.
The DOSs of m- and o-M3N4 (M=Si, Ge, Sn) are plotted in Figs. 2 and 3, respectively. Like Fig. 1, TDOSs drawn in Figs. 2(a) and 3(a) look similar on the gross scale. The reasons may be (i) the same crystal structure; (ii) Ge and Sn are iso-electronic elements. A subtle difference in the CB peaks can be noticed, which is due to the different atomic contributions in Si3N4, Ge3N4, and Sn3N4. For example, PDOSs of Ge/Sn at different coordinations are somehow different in the peak intensities indicating the difference of local environment. The sharp VB edge of M3N4(M=Si, Ge, Sn) implies that a lot of electrons may be distributed there. The valence band maxima are dominated by the p-orbitals of nitrogen atoms.[36] The monoclinic phases plotted in Figs. 2(b),2(c), and 2(d) show some differences, which are related to the atomic coordination and lattice constant (hence the bond length). As shown in Figs. 3(b),3(c), and 3(d), the DOSs of orthorhombic phases are much higher than the DOSs of tetragonal and monoclinic phases. Comparing with Ge3N4 and Si3N4, Sn3N4 has a steeper VB edge, a smaller band gap, and a less sharp CB edge. The interactions among different orbitals demonstrate the strong covalent Si/Ge/Sn–N bonding. The VB band width obeys the rule
. The bonding interactions between M and N atoms play an important role in the stabilities of M3N4 (M=Si, Ge, Sn).
3.3. Thermodynamic propertiesThe isobaric heat capacity
is an important parameter indicating the loss ability and heat retention. Although the constant-volume heat capacity
gives direct information of anharmonic effects, it is
that is experimentally measured, and it contains both the anharmonic effect and quasi-harmonic contribution leading to its departure from the Dulong–Petit’s limit (
for monoatomic materials).[37] In Fig. 4, one can see that the heat capacities of t-, m-, and o-M3N4 (M=Si, Ge, Sn) are quite similar in appearance. From 0 K to 300 K, the heat capacities follow the Debye law (
and vanish when the temperature tends to zero. Then,
gradually reaches an almost linear region (300 K
) and the increasing trend becomes gentler. When
,
increases monotonically with the rising of temperature (due to anharmonic effects), which is common to all solids at high temperatures. As shown in Fig. 4(a), the temperature effect on the heat capacity of t- and m-Si3N4 is more significant than that on o-Si3N4. In Fig. 4(b), the three curves show that
dependences on temperature are quite similar. For Sn3N4, the tetragonal and monoclinic phases have larger heat capacities than the orthorhombic phase. The knee points (at 200 K) shown in Fig. 4(c) also demonstrate that
of Sn3N4 increases more rapidly than the curves of Si3N4 and Ge3N4.
Microscopic entropy can be defined as a measure of the disorder of a compound. As expected, this parameter is highly dependent on the temperature T. Figure 5 shows the zero-pressure entropy S versus temperature for Si3N4, Ge3N4, and Sn3N4. As a non-purely-anharmonic quantity, one could not expect the linear relationship between S and T. It is clearly seen that S nearly keeps unchanged when
. In the temperature range of 50 K
, the entropies are variable by power exponent with the temperature increasing. At higher temperatures, S increases almost linearly with enhanced temperature. As shown in Fig. 5(a), the curves are essentially indistinguishable below 100 K. The highest entropy is m-Si3N4, followed by t-Si3N4 and o-Si3N4. In Fig. 5(b), the discrepancy of S increases with applied temperature. As plotted in Fig. 5(c), the entropies of t-Sn3N4 and m-Sn3N4 are nearly the same in the whole temperature range of 0 K
. It is worthy of note that the entropies of Si3N4 are much lower than those of Ge3N4 and Sn3N4. The results show that Sn3N4 has the lowest entropy. The effect of temperature on S is more significant at high temperature than that at low temperature. It is due mainly to the fact that as the temperature increases the lattice atoms vibrate more vigorously, inducing an increase in entropy.
The normalized vibrational internal energy
is the zero-temperature vibrational internal energy) is plotted in Fig. 6(a). One can see that
increases with temperature for all compounds (especially at
), and gradually tends to the linear increases at higher temperatures, but the slopes are different. The nine curves coincide at 0 K and diverge more and more when the temperature increases. These
curves are almost unchanged in the temperature range of 0 K
. Our results show a rather dramatic increase in
(t-Sn3N4) with T until the temperature tends to 1000 K. At a fixed temperature, m-Sn3N4 has a slightly lower
than t-Sn3N4. Finally, the increasing trend of
while going from Sn3N4 (fastest), to Ge3N4 (moderate) and to Si3N4 (slowest). The Grüneisen parameter γ, which describes the alteration in the lattice vibrations, can predict the anharmonic properties of crystals, such as the temperature-dependent Debye frequency and cell volume. Consequently, the Grüneisen parameters of t-, m-, and o-M3N4 (M=Si, Ge, Sn) are drawn in Fig. 6(b).
As shown in Fig. 6(b), the Grüneisen parameter
increases slightly with the temperature increasing, indicating that the anharmonicities of these compounds are not closely related to the temperature. The temperature effect on
is not as significant as that on
and S. In the temperature range of 0 K
,
of t-Sn3N4 decreases by 1.27%, while
of t-Si3N4, m-Si3N4, o-Si3N4, t-Ge3N4, m-Ge3N4, o-Ge3N4, m-Sn3N4, and o-Sn3N4 increase by 1.71%, 1.63%, 0.35%, 0.59%, 3.68%, 0.14%, 5.34%, and 1.07%, respectively. The ratio of orthorhombic phase is highest followed by the monoclinic phase, the tetragonal phase possesses the least. The effect of temperature is more pronounced for m-Sn3N4. The predicted heat capacity
(
) for t-Si3N4, m-Si3N4, o-Si3N4, t-Ge3N4, m-Ge3N4, o-Ge3N4, t-Sn3N4, m-Sn3N4, and o-Sn3N4 are
and
(
), respectively. Note that we could not find any other results for our comparison. Hence our calculation can serve as a prediction for future study of the thermal properties of
M3N
4 (
M=Si, Ge, Sn).
3.4. Optical propertiesOptical properties of solids are defined by the complex dielectric function
, where
is the real part of the dielectric function and
is the imaginary part of the dielectric function. The real part is determined from the imaginary part using the Kramers–Krönig relations.[38]
means that light cannot spread the material in the corresponding energy region. It can be seen that the spectrum shapes of the real part of Ge3N4/Sn3N4, which are shown in Figs. 7(a),7(b), and 7(c), are similar. During the photon energy increased process,
reaches the highest values of 7.26 (at 7.05 eV), 6.48 (at 6.07 eV), 8.82 (at 6.22 eV), 8.35 (at 5.47 eV), 7.27 (at 4.24 eV), 10.28 (at 2.95 eV), 8.81 (at 2.65 eV), 9.54 (at 0 eV), and 10.85 (at 2.06 eV), and then decreases to the valleys of −2.23 (at 14.82 eV), −1.48 (at 14.71 eV), −3.27 (at 13.86 eV), −2.04 (at 13.57 eV), −1.30 (at 13.36 eV), −2.77 (at 12.04 eV), −1.54 (at 11.17 eV), −1.24 (at 10.79 eV), and −2.24 (at 10.72 eV) for t-Si3N4, t-Ge3N4, t-Sn3N4, m-Si3N4, m-Ge3N4, m-Sn3N4, o-Si3N4, o-Ge3N4, and o-Sn3N4, respectively. The static dielectric constant
is found to be 4.47, 4.24, 5.71, 5.59, 5.30, 8.13, 7.69, 9.54, and 10.57 for t-Si3N4, t-Ge3N4, t-Sn3N4, m-Si3N4, m-Ge3N4, m-Sn3N4, o-Si3N4, o-Ge3N4, and o-Sn3N4, respectively.
of t-, m-, and o-Si3N4 are much smaller than those of elemental semiconductors Si and Ge, but are larger than
-,
- and
-Si3N4.[39] m- and t-Sn3N4 have the highest dielectric constants than any other compound; hence they can be used as candidates for dense refractive materials.
The imaginary dielectric function
represents the optical absorption of solids. Large difference of the dielectric functions
can be seen in Fig. 7, where the peaks are obviously shifted in different compounds. It is clearly observed in Figs. 7(d) and 7(e) that there are two peaks in the imaginary part of t- and m-Si3N4, the main peaks of t-Sn3N4(m-Si3N4) are located at 4.69 (10.31) and 8.44 (12.56) eV. Each spectrum shown in Fig. 7(f) has a prominent absorption peak. The three main peaks at 4.99, 6.21, and 9.51 eV for o-Si3N4, o-Ge3N4, and o-Sn3N4 have also been revealed, respectively. The amplitude of the main peak in the orthorhombic phase is significantly higher than the peaks in the other two phases. The maxima of
and
for the nine compounds have shifted to the low energy region following the sequence
. In Fig. 7(d), the rising curves show that the first optical point arises at 3.6 (2.2/0) eV for t-Si3N4 (t-Ge3N4/t-Sn3N4). This is due to the fact that the number of points contributing toward
is enhanced abruptly. It is also noted that the spectral shape of m-Si3N4 is little dissimilar from the other compounds, demonstrating somehow the different bonding nature of this system.
The reflectivity
of different nitrides is plotted in Figs. 8(a),8(b), and 8(c). As deducible from the dielectric function, up to the sharp plasma edge, all main peaks cause from inter-band transitions and for higher energies, the reflectivity vanishes. The maximum
of the above-mentioned nitrides appears in the energy range between 10 eV and 25 eV. As shown in Fig. 8(a), the
begins to increase from a relatively small value (12.71% for t-Si3N4, 16.52% for t-Ge3N4, and 22.10% for t-Sn3N4) to attain a maximum value (50.22% for photon energy at 17.11eV for t-Si3N4, 48.90% for photon energy at 14.21 eV for t-Ge3N4, and 43.62% for photon energy at 12.45 eV for t-Sn3N4). Then, the
decreases with the increasing photon energy. In Fig. 8(b), the curves do not exceed 40% through the whole energy range of 0 eV
, indicating that the monoclinic M3N4 (M=Si, Ge, Sn) are transparent materials. These nitrides may find applications as transparent coatings in the visible light region (1.6 eV
). For the orthorhombic phase shown in Fig. 8(c), the
is higher than 50% in the energy range of 12 eV
, indicating that photons in this energy region cannot travel through the solid. The sharp reflectivity plasma edges are located at 19.5, 23, and 25 eV for o-Si3N4, o-Ge3N4, and o-Sn3N4, respectively.
The photon energy dependences of energy loss function
are plotted in Figs. 8(d),8(e), and 8(f).
describes the energy-loss of a fast electron traversing in the material.[40] The peak of
describes the characteristic of plasma oscillation, the frequency of collective oscillation of the valence electrons. The dominant peaks in the loss function, which are due mainly to a longitudinal oscillation of the valence electrons as whole against the cores, have been found to locate at approximately 19.0, 22.2, and 24.5 eV for Si3N4, Ge3N4, and Sn3N4, respectively. These peaks correspond to the rapid reduction of
. In Figs. 8(d) and 8(e), the main peaks are broad, which are due mainly to the contributions of N-2p to M-p transitions. Furthermore, some small peaks are observed in t-Ge3N4, t-Sn3N4, and m-Ge3N4 which indicate weak resonances. It can be seen in Fig. 8(f) that the orthorhombic phases have enhanced amplitudes in
compared with the tetragonal and monoclinic phases. Unfortunately, no optical data are available in the literature for our comparison.
